3.4.77 \(\int \frac {x (A+B x)}{(a+c x^2)^{5/2}} \, dx\)

Optimal. Leaf size=50 \[ \frac {-A-B x}{3 c \left (a+c x^2\right )^{3/2}}+\frac {B x}{3 a c \sqrt {a+c x^2}} \]

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Rubi [A]  time = 0.01, antiderivative size = 47, normalized size of antiderivative = 0.94, number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {778, 191} \begin {gather*} \frac {B x}{3 a c \sqrt {a+c x^2}}-\frac {A+B x}{3 c \left (a+c x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x*(A + B*x))/(a + c*x^2)^(5/2),x]

[Out]

-(A + B*x)/(3*c*(a + c*x^2)^(3/2)) + (B*x)/(3*a*c*Sqrt[a + c*x^2])

Rule 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rule 778

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((a*(e*f + d*g) -
(c*d*f - a*e*g)*x)*(a + c*x^2)^(p + 1))/(2*a*c*(p + 1)), x] - Dist[(a*e*g - c*d*f*(2*p + 3))/(2*a*c*(p + 1)),
Int[(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && LtQ[p, -1]

Rubi steps

\begin {align*} \int \frac {x (A+B x)}{\left (a+c x^2\right )^{5/2}} \, dx &=-\frac {A+B x}{3 c \left (a+c x^2\right )^{3/2}}+\frac {B \int \frac {1}{\left (a+c x^2\right )^{3/2}} \, dx}{3 c}\\ &=-\frac {A+B x}{3 c \left (a+c x^2\right )^{3/2}}+\frac {B x}{3 a c \sqrt {a+c x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 32, normalized size = 0.64 \begin {gather*} \frac {B c x^3-a A}{3 a c \left (a+c x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x*(A + B*x))/(a + c*x^2)^(5/2),x]

[Out]

(-(a*A) + B*c*x^3)/(3*a*c*(a + c*x^2)^(3/2))

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IntegrateAlgebraic [A]  time = 0.42, size = 32, normalized size = 0.64 \begin {gather*} \frac {B c x^3-a A}{3 a c \left (a+c x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(x*(A + B*x))/(a + c*x^2)^(5/2),x]

[Out]

(-(a*A) + B*c*x^3)/(3*a*c*(a + c*x^2)^(3/2))

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fricas [A]  time = 0.43, size = 49, normalized size = 0.98 \begin {gather*} \frac {{\left (B c x^{3} - A a\right )} \sqrt {c x^{2} + a}}{3 \, {\left (a c^{3} x^{4} + 2 \, a^{2} c^{2} x^{2} + a^{3} c\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(B*x+A)/(c*x^2+a)^(5/2),x, algorithm="fricas")

[Out]

1/3*(B*c*x^3 - A*a)*sqrt(c*x^2 + a)/(a*c^3*x^4 + 2*a^2*c^2*x^2 + a^3*c)

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giac [A]  time = 0.24, size = 26, normalized size = 0.52 \begin {gather*} \frac {\frac {B x^{3}}{a} - \frac {A}{c}}{3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(B*x+A)/(c*x^2+a)^(5/2),x, algorithm="giac")

[Out]

1/3*(B*x^3/a - A/c)/(c*x^2 + a)^(3/2)

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maple [A]  time = 0.05, size = 29, normalized size = 0.58 \begin {gather*} -\frac {-B c \,x^{3}+a A}{3 \left (c \,x^{2}+a \right )^{\frac {3}{2}} a c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(B*x+A)/(c*x^2+a)^(5/2),x)

[Out]

-1/3*(-B*c*x^3+A*a)/(c*x^2+a)^(3/2)/a/c

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maxima [A]  time = 0.55, size = 51, normalized size = 1.02 \begin {gather*} -\frac {B x}{3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} c} + \frac {B x}{3 \, \sqrt {c x^{2} + a} a c} - \frac {A}{3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(B*x+A)/(c*x^2+a)^(5/2),x, algorithm="maxima")

[Out]

-1/3*B*x/((c*x^2 + a)^(3/2)*c) + 1/3*B*x/(sqrt(c*x^2 + a)*a*c) - 1/3*A/((c*x^2 + a)^(3/2)*c)

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mupad [B]  time = 1.09, size = 34, normalized size = 0.68 \begin {gather*} \frac {B\,x^3}{3\,a\,{\left (c\,x^2+a\right )}^{3/2}}-\frac {A}{3\,c\,{\left (c\,x^2+a\right )}^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*(A + B*x))/(a + c*x^2)^(5/2),x)

[Out]

(B*x^3)/(3*a*(a + c*x^2)^(3/2)) - A/(3*c*(a + c*x^2)^(3/2))

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sympy [A]  time = 13.25, size = 95, normalized size = 1.90 \begin {gather*} A \left (\begin {cases} - \frac {1}{3 a c \sqrt {a + c x^{2}} + 3 c^{2} x^{2} \sqrt {a + c x^{2}}} & \text {for}\: c \neq 0 \\\frac {x^{2}}{2 a^{\frac {5}{2}}} & \text {otherwise} \end {cases}\right ) + \frac {B x^{3}}{3 a^{\frac {5}{2}} \sqrt {1 + \frac {c x^{2}}{a}} + 3 a^{\frac {3}{2}} c x^{2} \sqrt {1 + \frac {c x^{2}}{a}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(B*x+A)/(c*x**2+a)**(5/2),x)

[Out]

A*Piecewise((-1/(3*a*c*sqrt(a + c*x**2) + 3*c**2*x**2*sqrt(a + c*x**2)), Ne(c, 0)), (x**2/(2*a**(5/2)), True))
+ B*x**3/(3*a**(5/2)*sqrt(1 + c*x**2/a) + 3*a**(3/2)*c*x**2*sqrt(1 + c*x**2/a))

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